The 
-lollipop
graph is the graph obtained by joining a complete graph 
to a path
graph 
with a bridge (Gallian 2025). Precomputed properties
of lollipop graphs are available in the Wolfram
Language as GraphData[
"Lollipop", 
m, n
].
The 
-lollipop
graph is isomorphic to the paw graph. In general, the
-lollipop graph is isomorphic to
the 
-tadpole graph and the 
-lollipop is isomorphic to the (degenerate) 
-pineapple graph.
Special cases are summarized in the following table (where the names paw graph and banner graph appear in ISGCI).
 -lollipop graph | name |
 | paw graph |
 | hammer graph |
 |  -tadpole graph |
 |  -pineapple graph |
Lollipop graphs are geodetic.
